Double and Triple Integral Evaluator

Math Calculus • Multivariable Calculus

Written by STEM Calculators Team Published December 30, 2025 Updated February 24, 2026

Integral type: Coordinates: Integrand \(f\): Method: Order (2D): Order (3D):

Type constants like pi, e. Use sqrt(2), sin(x), x^2. The region graph is a 2D preview (for 3D it shows the x–y footprint and the z-range separately).

Region (2D):

Rectangle bounds

x min: x max: y min: y max:

(rectangle only)

Type I bounds

a (x min): b (x max): g₁(x): g₂(x):

For Type I, the natural order is \(dy\,dx\). If you want \(dx\,dy\), re-express the region as Type II.

Type II bounds

c (y min): d (y max): h₁(y): h₂(y):

For Type II, the natural order is \(dx\,dy\). If you want \(dy\,dx\), re-express the region as Type I.

Polar bounds (radians)

r min: r max: θ min: θ max:

Polar change of variables: \(x=r\cos\theta\), \(y=r\sin\theta\), and \(dA=r\,dr\,d\theta\). The tool evaluates \(\int\int f(r\cos\theta,r\sin\theta)\,r\,dr\,d\theta\).

Box bounds (3D)

x min: x max: y min: y max: z min: z max:

Numeric accuracy (optional)

Tolerance (ε): Max recursion: Preview samples (graph): Show grid:

Ready

Enter an integrand and bounds, then click “Calculate”.

Region graph

Region preview (x–y plane)

Axes are in your input units. Hover to see \((x,y)\) and \(f(x,y)\) (at a representative z for 3D).

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Theory

Double and triple integrals as iterated integrals

A double integral accumulates a function over a region \(R\subset\mathbb{R}^2\):

\[ \iint_R f(x,y)\,dA \]

A triple integral accumulates a function over a volume \(V\subset\mathbb{R}^3\):

\[ \iiint_V f(x,y,z)\,dV \]

Rectangles / boxes (constant bounds)

For a rectangle \(R=[a,b]\times[c,d]\), we can integrate in either order:

\[ \iint_R f\,dA =\int_{y=c}^{d}\int_{x=a}^{b} f(x,y)\,dx\,dy =\int_{x=a}^{b}\int_{y=c}^{d} f(x,y)\,dy\,dx \]

The tool’s “Switch order” option recomputes the same value using the other iterated form (for rectangles).

Non-rectangular regions (Type I / Type II)

Many regions are described by variable-dependent bounds:

Type I (vertical slices): \(a\le x\le b\), \(g_1(x)\le y\le g_2(x)\), \[ \iint_R f\,dA=\int_{x=a}^{b}\int_{y=g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx \]
Type II (horizontal slices): \(c\le y\le d\), \(h_1(y)\le x\le h_2(y)\), \[ \iint_R f\,dA=\int_{y=c}^{d}\int_{x=h_1(y)}^{h_2(y)} f(x,y)\,dx\,dy \]

Switching order for non-rectangular regions usually requires rewriting the region in the other form.

Change of variables (polar coordinates)

In polar coordinates \(x=r\cos\theta\), \(y=r\sin\theta\). The area element becomes:

\[ dA = r\,dr\,d\theta \]

So a polar integral typically looks like:

\[ \iint_R f(x,y)\,dA =\int_{\theta=\alpha}^{\beta}\int_{r=r_1}^{r_2} f(r\cos\theta,r\sin\theta)\,r\,dr\,d\theta \]

The factor \(r\) is the Jacobian for the polar transformation.

Triple integrals as iterated integrals

For a box \(V=[a,b]\times[c,d]\times[p,q]\), you can integrate in any order:

\[ \iiint_V f\,dV=\int\int\int f(x,y,z)\,dx\,dy\,dz \]

Changing the order changes the intermediate steps, but (when all integrals exist) the final value is the same.

Worked example

Evaluate \(\iint_R x y\,dx\,dy\) over \(R=[0,1]\times[0,2]\):

\[ \begin{aligned} \int_{0}^{2}\int_{0}^{1} x y\,dx\,dy &=\int_{0}^{2}\left(y\int_{0}^{1} x\,dx\right)\,dy \\ &=\int_{0}^{2}\left(y\cdot \frac{x^2}{2}\Big|_{0}^{1}\right)\,dy \\ &=\frac{1}{2}\int_{0}^{2} y\,dy =\frac{1}{2}\cdot \frac{y^2}{2}\Big|_{0}^{2} =1 \end{aligned} \]

Notes on “symbolic vs numeric”

Exact (symbolic) evaluation is only guaranteed for certain families of functions (especially polynomials). For more complicated integrands or regions, numerical integration is used.

Exact: best for polynomials with constant bounds.
Numeric: robust; uses adaptive methods with a tolerance \(\varepsilon\).

If the function is undefined on part of the region (e.g., \(\sqrt{x}\) for \(x<0\)), the integral may fail or be invalid.

Frequently Asked Questions

What is the difference between a double integral and a triple integral?

A double integral accumulates a quantity over a region in the x-y plane, while a triple integral accumulates over a 3D region in x-y-z space. The main difference is the number of variables integrated and the number of bounds you must provide.

How do I choose the order of integration for an iterated integral?

The order is determined by which variable is integrated first, second, and (for triple integrals) third. Enter bounds in the same sequence as the integration order you intend to evaluate.

Why can changing the integration order affect the difficulty of the integral?

Different orders can lead to intermediate expressions that are easier or harder to integrate. While the value of a well-behaved definite integral is the same for valid equivalent bounds, the algebra and antiderivatives can vary significantly.

When does a double or triple integral represent volume or mass?

If the integrand is 1, the integral gives area (double) or volume (triple) over the region. If the integrand is a density function, the integral gives total mass or total amount over that region.

Region graph

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